Recent photometric observations by the Hubble Space Telescope have revealed the physical properties of stellar galactic nuclei in nucleated dwarf galaxies in the Virgo cluster of galaxies. In order to elucidate the formation processes of nucleated dwarfs, we numerically investigate gasdynamics, star formation, and chemical evolution within the central 1 kpc of gas disks embedded within the galactic stellar components of nonnucleated dwarfs. We find that high-density, compact stellar systems can be formed in the central regions of dwarfs as a result of dissipative, repeated merging of massive stellar and gaseous clumps developed from nuclear gaseous spiral arms as a consequence of local gravitational instability. The central stellar components are found to have stellar masses that are typically 5% of their host dwarfs' and show very flattened shapes, rotational kinematics, and central velocity dispersions significantly smaller than those of their host dwarfs. We also find that more massive dwarfs can develop more massive, more metal-rich, and higher density stellar systems in their central regions, because star formation and chemical enrichment proceed more efficiently owing to the less dramatic suppression of star formation by supernova feedback effects in more massive dwarfs. Based on these results, we suggest that gas-rich, nonnucleated dwarfs can be transformed into nucleated ones as a result of dissipative gasdynamics in their central regions. We discuss the origin of the observed correlations between physical properties of stellar galactic nuclei and those of their host galaxies.